There exist constants $c_1$ and $c_2$ such that
\[c_1 \begin{pmatrix} 2 \\ 3 \end{pmatrix} + c_2 \begin{pmatrix} -2 \\ 5 \end{pmatrix} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}.\]Enter the ordered pair $(c_1,c_2).$
From the given equation, $2c_1 - 2c_2 = -1$ and $3c_1 + 5c_2 = 4.$  Solving, we find
\[(c_1,c_2) = \boxed{\left( \frac{3}{16}, \frac{11}{16} \right)}.\]